Matrix Products: Sizes of A & B When AB & BA Are Defined

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Homework Help Overview

The discussion revolves around the conditions under which the matrix products AB and BA are defined, specifically focusing on the sizes of matrices A and B. Participants are exploring whether A and B must be square matrices and the implications of their dimensions on the defined products.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the relationships between the dimensions of matrices A and B necessary for the products AB and BA to be defined. There is an exploration of whether the matrices must be square and what specific size relationships must hold for the products to exist.

Discussion Status

Some participants have provided insights into the necessary conditions for the products to be defined, noting specific relationships between the dimensions of the matrices. There appears to be a productive exploration of the topic, with multiple interpretations being discussed.

Contextual Notes

Participants are considering the implications of matrix dimensions and the definitions of matrix multiplication, including the potential constraints of homework rules regarding the exploration of these concepts.

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Homework Statement


What do we know about the sizes of the matrices A and
B if both of the products AB and BA are defined?


Homework Equations





The Attempt at a Solution


I determined the sizes, of which were correct; but I was wondering if A and B would always be squares. Well, would they always be squares?
 
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Well, when is the product AB defined? How about BA?
 
Bashyboy said:

Homework Statement


What do we know about the sizes of the matrices A and
B if both of the products AB and BA are defined?


Homework Equations





The Attempt at a Solution


I determined the sizes, of which were correct; but I was wondering if A and B would always be squares. Well, would they always be squares?

Suppose A is m X n (m rows and n columns), and B is r X s. For AB to be defined, what relationship must there be between A's rows or columns and B's rows or columns? How many rows and columns will AB have?

For BA to be defined, what relationship must there be between B's rows or columns and A's rows or columns? How many rows and columns will BA have?
 
For AB to be defined, n = r. For BA to be defined, s = m. Right?
 
Bashyboy said:
For AB to be defined, n = r. For BA to be defined, s = m. Right?
Right.
 

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